The lepton flavour violating (LFV) reactions {\gamma \gamma \to \ell \ell '} ({\ell ,\ell ^'= e, \mu ,\tau }, \ell \not =\ell ') which arise at the one loop order of perturbation theory are studied at energies of interest for the \gamma \gamma option of the future ILC. The LFV mechanism is provided by low energy R-conserving SUSY with non diagonal slepton mass matrices. The average slepton masses {\mathaccent "0365\relax {m}} and the off diagonal matrix elements \Delta m are treated as model independent free phenomenological parameters in order to discover regions in the parameter space, where the signal cross section may be observable, comparing to the existing bounds on these parameters provided by the non-observation of radiative LFV decays and discuss how to reduce the standard model background.

PACS numbers: 11.30.Fs, 11.30.Pb, 12.60.Jv, 14.80.Ly